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Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
The area of the region bounded by the curve \(y=f(x)\), the \(X\)-axis and the lines \(x=a\) and \(x=b\), where \(-\infty < a < b < -2\), is

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Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that \(A\) is either symmetric or skew-symmetric or both, and det \((A)\) is divisible by \(p\) is

Considering only the principal values of the inverse trigonometric functions, the value of $\frac{3}{2}{\cos }^{-1}\sqrt{\frac{2}{2+{\pi }^2}}+\frac{1}{4}{\sin }^{-1}\frac{2\sqrt{2}\pi }{2+{\pi }^2}+{\tan }^{-1}\frac{\sqrt{2}}{\pi }$ is _______. (If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places)  

Consider the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let \(S(p, q)\) be a point in the first quadrant such that \(\frac{p^2}{9}+\frac{q^2}{4}>1\). Two tangents are drawn from \(S\) to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point \(T\) in the fourth quadrant. Let \(R\) be the vertex of the ellipse with positive \(x\)-coordinate and \(O\) be the center of the ellipse. If the area of the triangle \(\triangle O R T\) is \(\frac{3}{2}\), then which of the following options is correct?

Let $\text{z}=\frac{-1+\sqrt{3}\text{i}}{2}$ , where $\text{i}=\sqrt{-1}$ , and $\text{r},\text{s}\in \left\{1,2,3\right\}.$ Let $\text{P}=\left[\begin{array}{cc}{\left(-\mathrm{z}\right)}^{\text{r}}& {\text{z}}^{2\text{s}}\\ {\text{z}}^{2\text{s}}& {\text{z}}^{\text{r}}\end{array}\right]$ and $\text{I}$ be the identity matrix of order 2. Then the total number of ordered pairs $(\text{r},\text{s})$ for which ${\text{P}}^2=-\text{I}$ is

Let $M$ be a $3\times 3$ invertible matrix with real entries and let $I$ denote the $3\times 3$ identity matrix. $M^{-1}=adj\left(adjM\right)$, then which of the following statements is/are ALWAYS TRUE?

Let $f :\left(0, \infty \right)\to R$ be given by $f\left(x\right)= \underset{\frac{1}{x}}{\overset{x}{\int }}{e}^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t},$ then

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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
When the gas bubble is at height \(y\) from the bottom, its temperature is

Consider the system of equations \(x-2 y+3 z=-1, x-3 y+4 z=1\) and \(-x+y-2 z=k\)
Statement 1 The system of equations has no solution for \(k \neq 3\).
Statement 2 The determinant \(\left|\begin{array}{ccc}1 & 3 & -1 \\ -1 & -2 & k \\ 1 & 4 & 1\end{array}\right| \neq 0\), for \(k \neq 3\).

A parallel plate capacitor \(C\) with plates of unit area and separation \(d\) is filled with a liquid of dielectric constant \(K=2\). The level of liquid is \(\frac{d}{3}\) initially. Suppose the liquid level decreases at a constant speed \(v\), the time constant as a function of time \(t\) is

A man walks a distance of 3 units from the origin towards the North-East \(\left(\mathrm{N} 45^{\circ} \mathrm{E}\right)\) direction. From there, he walks a distance of 4 units towards the North-West ( \(\mathrm{N} 45^{\circ} \mathrm{W}\) ) direction to reach a point \(P\). Then, the position of \(P\) in the Argand plane is

The value of ${\left({\left({\log }_{2}9\right)}^2\right)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is_____.

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In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The volume of this \(\mathrm{HCP}\) unit cell is